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April 7th, 2013, 06:37 PM   #1
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Convergence and divergence

1.

infinity
E.........1/n
n = 1

2.

infinity
E ........1/n^2
n = 1

The first series diverges and the second converges to pi^2/6. Somewhere between the two series, behaviour switches from divergence to convergence. Where is the dividing line?

First, I don't get why the first series diverges and the second series converges. They are both approaching 0 but the second series is approaching it faster. Why is this so?
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April 8th, 2013, 12:56 AM   #2
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Re: Convergence and divergence

Quote:
Originally Posted by math221
Where is the dividing line?
The question is somewhat vague. Depending on what you want will give you different answers. For example, we could say



Is the dividing line.

Quote:
Originally Posted by math221
I don't get why the first series diverges and the second series converges.
The first series is just not fast enough to converge. Furthermore, taking the set of primes instead of the integers diverges as O(log(log n)) forcing the more general sum to diverge (like log(n)).

Quote:
Originally Posted by math221
They are both approaching 0 but the second series is approaching it faster.
The limit test is a necessary but not sufficient to test convergence.
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April 8th, 2013, 03:41 AM   #3
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Re: Convergence and divergence

[color=#000000] is divergent because .

is convergent because .[/color]
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April 8th, 2013, 07:39 AM   #4
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Re: Convergence and divergence

By the ratio test, converges as long as
so that "1" is the "dividing line".
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